Integrand size = 12, antiderivative size = 97 \[ \int \frac {\cos (b x) \operatorname {CosIntegral}(b x)}{x^3} \, dx=-\frac {\cos ^2(b x)}{4 x^2}-\frac {\cos (b x) \operatorname {CosIntegral}(b x)}{2 x^2}-\frac {1}{4} b^2 \operatorname {CosIntegral}(b x)^2-b^2 \operatorname {CosIntegral}(2 b x)+\frac {b \cos (b x) \sin (b x)}{2 x}+\frac {b \operatorname {CosIntegral}(b x) \sin (b x)}{2 x}+\frac {b \sin (2 b x)}{4 x} \]
-1/4*b^2*Ci(b*x)^2-b^2*Ci(2*b*x)-1/2*Ci(b*x)*cos(b*x)/x^2-1/4*cos(b*x)^2/x ^2+1/2*b*Ci(b*x)*sin(b*x)/x+1/2*b*cos(b*x)*sin(b*x)/x+1/4*b*sin(2*b*x)/x
Time = 0.01 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00 \[ \int \frac {\cos (b x) \operatorname {CosIntegral}(b x)}{x^3} \, dx=-\frac {\cos ^2(b x)}{4 x^2}-\frac {\cos (b x) \operatorname {CosIntegral}(b x)}{2 x^2}-\frac {1}{4} b^2 \operatorname {CosIntegral}(b x)^2-b^2 \operatorname {CosIntegral}(2 b x)+\frac {b \cos (b x) \sin (b x)}{2 x}+\frac {b \operatorname {CosIntegral}(b x) \sin (b x)}{2 x}+\frac {b \sin (2 b x)}{4 x} \]
-1/4*Cos[b*x]^2/x^2 - (Cos[b*x]*CosIntegral[b*x])/(2*x^2) - (b^2*CosIntegr al[b*x]^2)/4 - b^2*CosIntegral[2*b*x] + (b*Cos[b*x]*Sin[b*x])/(2*x) + (b*C osIntegral[b*x]*Sin[b*x])/(2*x) + (b*Sin[2*b*x])/(4*x)
Time = 1.04 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.31, number of steps used = 17, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.417, Rules used = {7070, 27, 3042, 3795, 14, 3042, 3793, 2009, 7076, 27, 4906, 27, 3042, 3778, 3042, 3783, 7237}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\operatorname {CosIntegral}(b x) \cos (b x)}{x^3} \, dx\) |
\(\Big \downarrow \) 7070 |
\(\displaystyle -\frac {1}{2} b \int \frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x^2}dx+\frac {1}{2} b \int \frac {\cos ^2(b x)}{b x^3}dx-\frac {\operatorname {CosIntegral}(b x) \cos (b x)}{2 x^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {1}{2} b \int \frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x^2}dx+\frac {1}{2} \int \frac {\cos ^2(b x)}{x^3}dx-\frac {\operatorname {CosIntegral}(b x) \cos (b x)}{2 x^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {1}{2} b \int \frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x^2}dx+\frac {1}{2} \int \frac {\sin \left (b x+\frac {\pi }{2}\right )^2}{x^3}dx-\frac {\operatorname {CosIntegral}(b x) \cos (b x)}{2 x^2}\) |
\(\Big \downarrow \) 3795 |
\(\displaystyle \frac {1}{2} \left (b^2 \int \frac {1}{x}dx-2 b^2 \int \frac {\cos ^2(b x)}{x}dx-\frac {\cos ^2(b x)}{2 x^2}+\frac {b \sin (b x) \cos (b x)}{x}\right )-\frac {1}{2} b \int \frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x^2}dx-\frac {\operatorname {CosIntegral}(b x) \cos (b x)}{2 x^2}\) |
\(\Big \downarrow \) 14 |
\(\displaystyle \frac {1}{2} \left (-2 b^2 \int \frac {\cos ^2(b x)}{x}dx+b^2 \log (x)-\frac {\cos ^2(b x)}{2 x^2}+\frac {b \sin (b x) \cos (b x)}{x}\right )-\frac {1}{2} b \int \frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x^2}dx-\frac {\operatorname {CosIntegral}(b x) \cos (b x)}{2 x^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{2} \left (-2 b^2 \int \frac {\sin \left (b x+\frac {\pi }{2}\right )^2}{x}dx+b^2 \log (x)-\frac {\cos ^2(b x)}{2 x^2}+\frac {b \sin (b x) \cos (b x)}{x}\right )-\frac {1}{2} b \int \frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x^2}dx-\frac {\operatorname {CosIntegral}(b x) \cos (b x)}{2 x^2}\) |
\(\Big \downarrow \) 3793 |
\(\displaystyle \frac {1}{2} \left (-2 b^2 \int \left (\frac {\cos (2 b x)}{2 x}+\frac {1}{2 x}\right )dx+b^2 \log (x)-\frac {\cos ^2(b x)}{2 x^2}+\frac {b \sin (b x) \cos (b x)}{x}\right )-\frac {1}{2} b \int \frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x^2}dx-\frac {\operatorname {CosIntegral}(b x) \cos (b x)}{2 x^2}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {1}{2} b \int \frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x^2}dx+\frac {1}{2} \left (-2 b^2 \left (\frac {\operatorname {CosIntegral}(2 b x)}{2}+\frac {\log (x)}{2}\right )+b^2 \log (x)-\frac {\cos ^2(b x)}{2 x^2}+\frac {b \sin (b x) \cos (b x)}{x}\right )-\frac {\operatorname {CosIntegral}(b x) \cos (b x)}{2 x^2}\) |
\(\Big \downarrow \) 7076 |
\(\displaystyle -\frac {1}{2} b \left (b \int \frac {\cos (b x) \operatorname {CosIntegral}(b x)}{x}dx+b \int \frac {\cos (b x) \sin (b x)}{b x^2}dx-\frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x}\right )+\frac {1}{2} \left (-2 b^2 \left (\frac {\operatorname {CosIntegral}(2 b x)}{2}+\frac {\log (x)}{2}\right )+b^2 \log (x)-\frac {\cos ^2(b x)}{2 x^2}+\frac {b \sin (b x) \cos (b x)}{x}\right )-\frac {\operatorname {CosIntegral}(b x) \cos (b x)}{2 x^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {1}{2} b \left (b \int \frac {\cos (b x) \operatorname {CosIntegral}(b x)}{x}dx+\int \frac {\cos (b x) \sin (b x)}{x^2}dx-\frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x}\right )+\frac {1}{2} \left (-2 b^2 \left (\frac {\operatorname {CosIntegral}(2 b x)}{2}+\frac {\log (x)}{2}\right )+b^2 \log (x)-\frac {\cos ^2(b x)}{2 x^2}+\frac {b \sin (b x) \cos (b x)}{x}\right )-\frac {\operatorname {CosIntegral}(b x) \cos (b x)}{2 x^2}\) |
\(\Big \downarrow \) 4906 |
\(\displaystyle -\frac {1}{2} b \left (b \int \frac {\cos (b x) \operatorname {CosIntegral}(b x)}{x}dx+\int \frac {\sin (2 b x)}{2 x^2}dx-\frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x}\right )+\frac {1}{2} \left (-2 b^2 \left (\frac {\operatorname {CosIntegral}(2 b x)}{2}+\frac {\log (x)}{2}\right )+b^2 \log (x)-\frac {\cos ^2(b x)}{2 x^2}+\frac {b \sin (b x) \cos (b x)}{x}\right )-\frac {\operatorname {CosIntegral}(b x) \cos (b x)}{2 x^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {1}{2} b \left (b \int \frac {\cos (b x) \operatorname {CosIntegral}(b x)}{x}dx+\frac {1}{2} \int \frac {\sin (2 b x)}{x^2}dx-\frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x}\right )+\frac {1}{2} \left (-2 b^2 \left (\frac {\operatorname {CosIntegral}(2 b x)}{2}+\frac {\log (x)}{2}\right )+b^2 \log (x)-\frac {\cos ^2(b x)}{2 x^2}+\frac {b \sin (b x) \cos (b x)}{x}\right )-\frac {\operatorname {CosIntegral}(b x) \cos (b x)}{2 x^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {1}{2} b \left (b \int \frac {\cos (b x) \operatorname {CosIntegral}(b x)}{x}dx+\frac {1}{2} \int \frac {\sin (2 b x)}{x^2}dx-\frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x}\right )+\frac {1}{2} \left (-2 b^2 \left (\frac {\operatorname {CosIntegral}(2 b x)}{2}+\frac {\log (x)}{2}\right )+b^2 \log (x)-\frac {\cos ^2(b x)}{2 x^2}+\frac {b \sin (b x) \cos (b x)}{x}\right )-\frac {\operatorname {CosIntegral}(b x) \cos (b x)}{2 x^2}\) |
\(\Big \downarrow \) 3778 |
\(\displaystyle -\frac {1}{2} b \left (b \int \frac {\cos (b x) \operatorname {CosIntegral}(b x)}{x}dx+\frac {1}{2} \left (2 b \int \frac {\cos (2 b x)}{x}dx-\frac {\sin (2 b x)}{x}\right )-\frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x}\right )+\frac {1}{2} \left (-2 b^2 \left (\frac {\operatorname {CosIntegral}(2 b x)}{2}+\frac {\log (x)}{2}\right )+b^2 \log (x)-\frac {\cos ^2(b x)}{2 x^2}+\frac {b \sin (b x) \cos (b x)}{x}\right )-\frac {\operatorname {CosIntegral}(b x) \cos (b x)}{2 x^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {1}{2} b \left (b \int \frac {\cos (b x) \operatorname {CosIntegral}(b x)}{x}dx+\frac {1}{2} \left (2 b \int \frac {\sin \left (2 b x+\frac {\pi }{2}\right )}{x}dx-\frac {\sin (2 b x)}{x}\right )-\frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x}\right )+\frac {1}{2} \left (-2 b^2 \left (\frac {\operatorname {CosIntegral}(2 b x)}{2}+\frac {\log (x)}{2}\right )+b^2 \log (x)-\frac {\cos ^2(b x)}{2 x^2}+\frac {b \sin (b x) \cos (b x)}{x}\right )-\frac {\operatorname {CosIntegral}(b x) \cos (b x)}{2 x^2}\) |
\(\Big \downarrow \) 3783 |
\(\displaystyle -\frac {1}{2} b \left (b \int \frac {\cos (b x) \operatorname {CosIntegral}(b x)}{x}dx-\frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x}+\frac {1}{2} \left (2 b \operatorname {CosIntegral}(2 b x)-\frac {\sin (2 b x)}{x}\right )\right )+\frac {1}{2} \left (-2 b^2 \left (\frac {\operatorname {CosIntegral}(2 b x)}{2}+\frac {\log (x)}{2}\right )+b^2 \log (x)-\frac {\cos ^2(b x)}{2 x^2}+\frac {b \sin (b x) \cos (b x)}{x}\right )-\frac {\operatorname {CosIntegral}(b x) \cos (b x)}{2 x^2}\) |
\(\Big \downarrow \) 7237 |
\(\displaystyle \frac {1}{2} \left (-2 b^2 \left (\frac {\operatorname {CosIntegral}(2 b x)}{2}+\frac {\log (x)}{2}\right )+b^2 \log (x)-\frac {\cos ^2(b x)}{2 x^2}+\frac {b \sin (b x) \cos (b x)}{x}\right )-\frac {\operatorname {CosIntegral}(b x) \cos (b x)}{2 x^2}-\frac {1}{2} b \left (\frac {1}{2} b \operatorname {CosIntegral}(b x)^2-\frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x}+\frac {1}{2} \left (2 b \operatorname {CosIntegral}(2 b x)-\frac {\sin (2 b x)}{x}\right )\right )\) |
-1/2*(Cos[b*x]*CosIntegral[b*x])/x^2 + (-1/2*Cos[b*x]^2/x^2 - 2*b^2*(CosIn tegral[2*b*x]/2 + Log[x]/2) + b^2*Log[x] + (b*Cos[b*x]*Sin[b*x])/x)/2 - (b *((b*CosIntegral[b*x]^2)/2 - (CosIntegral[b*x]*Sin[b*x])/x + (2*b*CosInteg ral[2*b*x] - Sin[2*b*x]/x)/2))/2
3.2.14.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(c + d*x)^(m + 1)*(Sin[e + f*x]/(d*(m + 1))), x] - Simp[f/(d*(m + 1)) Int[( c + d*x)^(m + 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && LtQ[m, - 1]
Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosInte gral[e - Pi/2 + f*x]/d, x] /; FreeQ[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - c*f, 0]
Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In t[ExpandTrigReduce[(c + d*x)^m, Sin[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f , m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1]))
Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbo l] :> Simp[(c + d*x)^(m + 1)*((b*Sin[e + f*x])^n/(d*(m + 1))), x] + (-Simp[ b*f*n*(c + d*x)^(m + 2)*Cos[e + f*x]*((b*Sin[e + f*x])^(n - 1)/(d^2*(m + 1) *(m + 2))), x] + Simp[b^2*f^2*n*((n - 1)/(d^2*(m + 1)*(m + 2))) Int[(c + d*x)^(m + 2)*(b*Sin[e + f*x])^(n - 2), x], x] - Simp[f^2*(n^2/(d^2*(m + 1)* (m + 2))) Int[(c + d*x)^(m + 2)*(b*Sin[e + f*x])^n, x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && LtQ[m, -2]
Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b _.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin[a + b*x ]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IG tQ[p, 0]
Int[Cos[(a_.) + (b_.)*(x_)]*CosIntegral[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)* (x_))^(m_.), x_Symbol] :> Simp[(e + f*x)^(m + 1)*Cos[a + b*x]*(CosIntegral[ c + d*x]/(f*(m + 1))), x] + (Simp[b/(f*(m + 1)) Int[(e + f*x)^(m + 1)*Sin [a + b*x]*CosIntegral[c + d*x], x], x] - Simp[d/(f*(m + 1)) Int[(e + f*x) ^(m + 1)*Cos[a + b*x]*(Cos[c + d*x]/(c + d*x)), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && ILtQ[m, -1]
Int[CosIntegral[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_)*Sin[(a_.) + ( b_.)*(x_)], x_Symbol] :> Simp[(e + f*x)^(m + 1)*Sin[a + b*x]*(CosIntegral[c + d*x]/(f*(m + 1))), x] + (-Simp[b/(f*(m + 1)) Int[(e + f*x)^(m + 1)*Cos [a + b*x]*CosIntegral[c + d*x], x], x] - Simp[d/(f*(m + 1)) Int[(e + f*x) ^(m + 1)*Sin[a + b*x]*(Cos[c + d*x]/(c + d*x)), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && ILtQ[m, -1]
Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Si mp[q*(y^(m + 1)/(m + 1)), x] /; !FalseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]
\[\int \frac {\operatorname {Ci}\left (b x \right ) \cos \left (b x \right )}{x^{3}}d x\]
\[ \int \frac {\cos (b x) \operatorname {CosIntegral}(b x)}{x^3} \, dx=\int { \frac {\cos \left (b x\right ) \operatorname {C}\left (b x\right )}{x^{3}} \,d x } \]
\[ \int \frac {\cos (b x) \operatorname {CosIntegral}(b x)}{x^3} \, dx=\int \frac {\cos {\left (b x \right )} \operatorname {Ci}{\left (b x \right )}}{x^{3}}\, dx \]
\[ \int \frac {\cos (b x) \operatorname {CosIntegral}(b x)}{x^3} \, dx=\int { \frac {\cos \left (b x\right ) \operatorname {C}\left (b x\right )}{x^{3}} \,d x } \]
\[ \int \frac {\cos (b x) \operatorname {CosIntegral}(b x)}{x^3} \, dx=\int { \frac {\cos \left (b x\right ) \operatorname {C}\left (b x\right )}{x^{3}} \,d x } \]
Timed out. \[ \int \frac {\cos (b x) \operatorname {CosIntegral}(b x)}{x^3} \, dx=\int \frac {\mathrm {cosint}\left (b\,x\right )\,\cos \left (b\,x\right )}{x^3} \,d x \]